A man sold a bicycle at 5% profit. If the cost had been 30% less and the selling price ₹ 63 less, he would have made a profit of 30%. What was the cost price of the bicycle?

Let the C.P. of the bicycle be ₹ $100$.

Profit % = 5%

As we know

$\text{Profit\% } = \dfrac{\text{Profit}}{\text{C.P.}} \times 100 \\[1em] \Rightarrow 5 = \dfrac{\text{Profit}}{100} \times 100 \\[1em] \Rightarrow 5 = \dfrac{\text{Profit}}{\cancel{100}} \times \cancel{100} \\[1em] \Rightarrow \text{Profit} = ₹ 5$

And

$\text{Profit = S.P. - C.P.}\\[1em] \Rightarrow 5 = \text{S.P.} - 100\\[1em] \Rightarrow \text{S.P.} = 100 + 5\\[1em] \Rightarrow \text{S.P.} = ₹ 105$

When C.P. of the bicycle is $30\%$ less.

$= 100 - \dfrac{30}{100} \times 100\\[1em] = 100 - \dfrac{30}{\cancel{100}} \times \cancel{100}\\[1em] = 100 - 30\\[1em] = ₹ 70$

∴ New C.P. = ₹ 70

New gain % = 30%.

As we know,

$\text{Gain\% } = \dfrac{\text{Gain}}{\text{C.P.}} \times 100 \\[1em] \Rightarrow 30 = \dfrac{\text{Gain}}{70} \times 100 \\[1em] \Rightarrow \text{Gain} = \dfrac{30 \times 70}{100}\\[1em] \Rightarrow \text{Gain} = \dfrac{2100}{100}\\[1em] \Rightarrow \text{Gain} = ₹ 21$

And

$\text{New Gain = New S.P. - New C.P.}\\[1em] \Rightarrow 21 = \text{New S.P.} - 70\\[1em] \Rightarrow \text{New S.P.} = 21 + 70\\[1em] \Rightarrow \text{New S.P.} = ₹ 91$

Difference in Selling Price = Original S.P. - New S.P.

= ₹ 105 - ₹ 91 = ₹ 14

∴ New S.P. is ₹ 14 less than original S.P.

Applying unitary method:

When sold for ₹ 14 less, the C.P. of the bicycle = ₹ 100.

When sold for ₹ 63 less, the C.P. of the bicycle = ₹ $\dfrac{100}{14} \times 63 = ₹ 450$.

Hence, cost price of the bicycle = ₹ 450.